多项式组的特征分解

任一多项式理想的特征对是指由该理想的约化字典序Grobner基G和含于其中的极小三角列C构成的有序对(G,C).当C为正则列或正规列时,分别称特征对(G,C)为正则的或正规的.当G生成的理想与C的饱和理想相同时,称特征对(G,C)为强的.一组多项式的(强)正则或(强)正规特征分解是指将该多项式组分解为有限多个(强)正则或(强)正规特征对,使其满足特定的零点与理想关系.本文简要回顾各种三角分解及相应零点与理想分解的理论和方法,然后重点介绍(强)正则与(强)正规特征对和特征分解的性质,说明三角列、Ritt特征列和字典序Grobner基之间的内在关联,建立特征对的正则化定理以及正则、正规特征对的强化方法,进而给出两种基于字典序Grobner基计算、按伪整除关系分裂和构建、商除可除理想等策略的(强)正规与(强)正则特征分解算法.这两种算法计算所得的强正规与强正则特征对和特征分解都具有良好的性质,且能为输入多元多项式组的零点提供两种不同的表示.本文还给出示例和部分实验结果,用以说明特征分解方法及其实用性和有效性.     

局部Artin主理想环上多项式理想的准素分解与根理想的计算

本文用极小Grobner基的标准型给出了局部Artin主理想环上单变元多项式理想的准素分解与根理想的计算.     

加细剖分样条函数空间的维数级数和基函数

在文[1]中,我们讨论了利用协调矩阵计算维数级数和基函数的Grobner基方法,本文考虑几种加细剖分样条函数空间的维数级数和基函数,给出了它们的表达式。 1 任意三角剖分的连续样条函数 任意三角剖分上连续样条函数空间的维数早已被确定。本节我们用Grobner基方法来计算其维数级数的发生函数和基函数。 设Δ是单连通区域D上的三角剖分,f_0~0(Δ)是Δ内点的个数,f_1~0(Δ)是内网线的个数,f_2~0(Δ)是三角形的个数,我们有     

准-Groebner基的计算

本文通过定义S－多项式，给出了系数环是整环的多项式中理想的准－Groebner基的一个算法，并据此给出了计算该理想极大无关变元组和维数的一种方法。     

K[X]^m中模的生成基及其应用

本文研究了多项式环上的素模中的生成基理论和方法。通过建立新的约化准则,得到了模中生成基的结构和机械计算方法。对于低维情形给出了素右生成基的充分必要条件。文中的方法本质地简化了传统的模中Grobner基方法。文中同时介绍了该方法在样条理论研究中的应用,并给出了一些计算例子。     

Grobner基的一个应用

固定一个项序,利用Buchberger算法求多项式环S=C[x1,x2,…,xn]上的理想I的Grbner基.根据S上任意多项式f（x1,x2,…,xn）用Grobner基表示时其余项唯一的特点,将其应用到求解多项式方程组问题.实例展示用Grobner基可证明一个联立方程式是无解的.     

循环差分-微分模上双变元维数多项式的Gr?bner基算法北大核心CSCD

Gr?bner基算法是在计算机辅助设计和机器人学、信息安全等领域广泛应用的重要工具.文章在周梦和Winkler(2008)给出的差分-微分模上Gr?bner基算法和差分-微分维数多项式算法基础上,进一步研究了分别差分部分和微分部分的双变元维数多项式算法.在循环差分-微分模情形,构造和证明了利用差分-微分模上Gr?bner基计算双变元维数多项式的算法.     

循环差分-微分模上双变元维数多项式的Gr?bner基算法

Gr?bner基算法是在计算机辅助设计和机器人学、信息安全等领域广泛应用的重要工具.文章在周梦和Winkler(2008)给出的差分-微分模上Gr?bner基算法和差分-微分维数多项式算法基础上,进一步研究了分别差分部分和微分部分的双变元维数多项式算法.在循环差分-微分模情形,构造和证明了利用差分-微分模上Gr?bner基计算双变元维数多项式的算法.     

多项式理想准素分解的算法

讨论了多项式理想准素分解的算法,对于理想坐标变换中关键的“一般位置”的计算,给出了一个确定算法,并建立了特别规范Groebner基的概念,利用这一概念及文中提供的相应算法,可以方便地计算理想的维数,进而使用降维的方法来得到高维理想的准素分解。     

可除的四元数代数上多项式环的Grobner基

设F是一个特征不等于2的域,A是,上的一个可除代数。本文研究了A上多项式环A[x1,X2,…,xn]中理想是有限生成的,以及它的Grobner基;也表明F[x1,x2,…,xn]中有限子集G是F[x1,x2,…,xn]的Griobner基当且仅当G是A[x1,x2,…,xn]中的Grobner基。     

Computing border bases

This paper presents several algorithms that compute border bases of a zero-dimensional ideal. The first relates to the FGLM algorithm as it uses a linear basis transformation. In particular, it is able to compute border bases that do not contain a reduced Gröbner basis. The second algorithm is based on a generic algorithm by Bernard Mourrain originally designed for computing an ideal basis that need not be a border basis. Our fully detailed algorithm computes a border basis of a zero-dimensional ideal from a given set of generators. To obtain concrete instructions we appeal to a degree-compatible term ordering σ and hence compute a border basis that contains the reduced σ-Gröbner basis. We show an example in which this computation actually has advantages over Buchberger's algorithm. Moreover, we formulate and prove two optimizations of the Border Basis Algorithm which reduce the dimensions of the linear algebra subproblems.     

诺特赋值环上Grbner基的性质

本文主要研究了诺特赋值环上多项式理想的Grbner基的性质.利用Buchberger算法,证明了约化Grbner基的存在性及当其首项系数为单位元时的唯一性.推广了极小Grbner基和约化Grbner基的概念.同时,我们给出了求极小Grbner基和约化Grbner基的算法.     

Gröbner bases of characteristic ideals of LRS over UFD

LetR be a unique factorization domain (UFD). A method of Gröbner bases and localization in commutative algebra is applied to compute and analyze the characteristic ideals of semi-infinite linear recurring sequences (lrs), infinite linear recurring sequences (LRS), and finite lrs over UFD. The canonical form of a minimal Gröbner basis of the homogeneous characteristic ideal is described for a finite segment of an lrs, from which a precise relation between every step in the classical Berlekamp-Massey algorithm and every member of the Gröbner basis is derived.     

On the stable set of associated prime ideals of a monomial ideal

It is shown that any set of nonzero monomial prime ideals can be realized as the stable set of associated prime ideals of a monomial ideal. Moreover, an algorithm is given to compute the stable set of associated prime ideals of a monomial ideal.     

Descentwise inexact proximal algorithms for smooth optimization

The proximal method is a standard regularization approach in optimization. Practical implementations of this algorithm require (i)?an algorithm to compute the proximal point, (ii)?a rule to stop this algorithm, (iii)?an update formula for the proximal parameter. In this work we focus on?(ii), when smoothness is present??so that Newton-like methods can be used for?(i): we aim at giving adequate stopping rules to reach overall efficiency of the method. Roughly speaking, usual rules consist in stopping inner iterations when the current iterate is close to the proximal point. By contrast, we use the standard paradigm of numerical optimization: the basis for our stopping test is a ??sufficient?? decrease of the objective function, namely a fraction of the ideal decrease. We establish convergence of the algorithm thus obtained and we illustrate it on some ill-conditioned problems. The experiments show that combining the proposed inexact proximal scheme with a standard smooth optimization algorithm improves the numerical behaviour of the latter for those ill-conditioned problems.     

The Neville-like form of the Fitzpatrick algorithm for rational interpolation

The Fitzpatrick algorithm, which seeks a Gr?bner basis for the solution of a system of polynomial congruences, can be applied to compute a rational interpolant. Based on the Fitzpatrick algorithm and the properties of an Hermite interpolation basis, we present a Neville-like algorithm for multivariate osculatory rational interpolation. It may be used to compute the values of osculatory rational interpolants at some points directly without computing the rational interpolation function explicitly.     

Generation of three-dimensional block-structured grids based on rotation of two-dimensional multiply connected cross sections

An algorithm for generating curvilinear block-structured grids in axisymmetric three-dimensional domains of any connectivity is developed. The organization of the connection between the blocks is automated. The grids constructed are used to compute ideal gas steady flows past axisymmetric bodies at a nonzero angle of attack.     

Multiplication Matrices and Ideals of Projective Dimension Zero

We introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and, in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero.     

Verifying exactness of relaxations for robust semi-definite programs by solving polynomial systems

In this paper, robust semi-definite programs are considered with the goal of verifying whether a particular LMI relaxation is exact. A procedure is presented showing that verifying exactness amounts to solving a polynomial system. The main contribution of the paper is a new algorithm to compute all isolated solutions of a system of polynomials. Standard techniques in computational algebra, often referred to as Stetter’s method [H.J. Stetter, Numerical Polynomial Algebra, SIAM, 2004], involve the computation of a Gröbner basis of the ideal generated by the polynomials and further require joint eigenvector computations in order to arrive at the zeros of the polynomial system. Our algorithm does neither require structural knowledge on the polynomial system, nor does it rely on the computation of joint eigenvectors.